Trace index and spectral flow in the entanglement spectrum of topological insulators

We investigate the entanglement spectra of topological insulators, which manifest edge states on a lattice with spatial boundaries. In the physical energy spectrum, a subset of the edge states that intersect the Fermi level translates to discontinuities in the trace of the single-particle entanglement spectrum, which we call a “trace index.” We find that any free-fermion topological insulator that exhibits spectral flow has a nonvanishing trace index, which provides us with a new description of topological invariants. In addition, we identify the signatures of spectral flow in the single-particle and many-body entanglement spectrum; in the process we present new methods to extract topological invariants and establish a connection between entanglement and quantum Hall physics.

Figure 1

(Color online) (a) Our chosen geometry is a square lattice with open boundary conditions along $\widehat{x}$ and periodic boundary conditions along $\widehat{y}$: an open-end hollow cylinder. Quantum Hall phases such as the Chern insulator exhibit modes that extend around the edges of the cylinder. Extended states on opposite edges have opposite chirality; the direction of propagation of these modes is indicated by red and blue arrows. (b) We perform a spatial entanglement cut by partitioning the cylinder into two halves; the left half is labeled $A$ and the right is $B$. For the Chern insulator, the spectrum of the reduced correlation matrix $G^A$ in region $A$ includes edge modes (green arrow) on the entanglement cut between $A$ and $B$. These edge modes contribute maximally to the entanglement entropy.

Figure 2

(Color online) (a) and (b), respectively, show the entanglement energy spectrum and the many-body entanglement spectrum (ES) of a trivial insulator that is adiabatically connected to the atomic insulator. The latter is scaled logarithmically. The many-body ES belongs in the half filled particle number sector, i.e., half the total number of single-particle entanglement eigenstates are occupied. (c) and (d), respectively, show the same spectra for a topological insulator that exhibits spectral flow. The many-body ES is plotted as a function of the total momenta $k_y$ of the many-body entanglement state. As a reference point, we define $k_y^{\mathit{GS}}$ as the total momentum of the entanglement ground state.

Figure 3

(Color online) (a)–(c) show, respectively, the physical energy spectrum, the single-particle entanglement spectrum (ES) and the trace of $G^A\left(k_y\right)$ for a trivial insulator ($C_1=0$) on an open-end cylinder. In (a) and (b), bulk bands on opposite sides of the gap are spectrally disconnected from each other. In (c), the trace of $G^A$ is a continuous function of $k_y$ so the trace index ${\mathcal{A}}_{U\left(1\right)}=0$. (d)–(f) show, respectively, the physical energy spectrum, the single-particle ES and the trace of $G^A\left(k_y\right)$ for a nontrivial insulator with $C_1=-1$. In (d) and (e), a set of edge states interpolate between bulk bands on opposite sides of the gap; this spectral flow distinguishes trivial and nontrivial Chern insulators. In (f), the trace of $G^A$ drops discontinuously by 1 at $k_y=0$; the trace index ${\mathcal{A}}_{U\left(1\right)}=-1$.

Figure 4

(Color online) (a), (c), (e), and (g) show schematically the physical energy spectrum, the single-particle entanglement spectrum (ES), the trace of the single-particle ES, and the entanglement energy spectrum, respectively, as functions of $k_y$ for a many-body ground state of a nontrivial Chern insulator. (b), (d), (f), and (h) label the new functions when a flux quantum is inserted through the symmetry axis of the cylinder. These are schematic figures that emphasize importance features in the physical and entanglement energy gaps; bulk bands away from the gaps are conveniently portrayed as dispersionless. Eigenvalues that are colored red, green, and blue correspond to eigenstates that extend over the left edge, the entanglement cut, and the right edge of the cylinder, respectively. Occupied single-particle states of a many-body Slater determinant are labeled by black circles in both the physical and entanglement energy spectra.

Figure 5

(Color online) Disorder-quenched trace index for the Chern insulator as a function of the disorder strength parameter $W$. The indices are plotted for cylinders of dimensions $N_x=10$ by $N_y=5$ (circles), $25\times 25$ (crosses), and $40\times 40$ (squares). We observe an increasingly sharp transition from the nontrivial insulator with (with trace index 1) to the trivial insulator (with trace index 0) as the dimension of the cylinder is increased.

Figure 6

(Color online) (a)–(c) show, respectively, the physical energy spectrum, the single-particle entanglement spectrum (ES), and the trace of $G^A\left(k_y\right)$ for a trivial insulator (${\chi }^{Z_2}$ even) on an open-end cylinder. (a) and (b) Bulk bands on opposite sides of the gap are spectrally disconnected from each other. (c) The trace of $G^A$ is a continuous function of $k_y$ so the trace index ${\mathcal{A}}_{Z_2}=0$. (d)–(f) show, respectively, the physical energy spectrum, the single-particle ES, and the trace of $G^A\left(k_y\right)$ for a nontrivial insulator with odd ${\chi }^{Z_2}$. (d) and (e) A set of edge states interpolate between bulk bands on opposite sides of the gap; this spectral flow distinguishes trivial and nontrivial $Z_2$ insulators. (f) The trace of $G^A$ jumps discontinuously by 1 in half the Brillouin zone $k_y\in [0,\pi ]$; the trace index ${\mathcal{A}}_{Z_2}=+1$.

Figure 7

(Color online) (a), (c), and (e) show schematically the physical energy spectrum, the single-particle entanglement spectrum (ES), and the trace of the single-particle ES, respectively, as functions of $k_y$, for a many-body ground state of a nontrivial QSHI. The subspace of occupied bands in the physical spectrum is closed under time-reversal. (b), (d), and (f) label the new functions when an integer charge-flux is inserted through the symmetry axis of the cylinder. For convenience, bulk bands away from the gaps are portrayed as dispersionless. Eigenvalues that are colored red, green, and blue correspond to eigenstates that extend over the left edge, the entanglement cut, and the right edge of the cylinder, respectively. Kramers’ doublets at $k_y=0$ and $\pi$ are labeled by thick dashes in the physical energy spectra; occupied single-particle states of a many-body Slater determinant are labeled by black circles.

Figure 8

(Color online) (a), (c), and (e) show schematically the physical energy spectrum, the single-particle entanglement spectrum (ES), and the entanglement energy spectrum, respectively, as functions of $k_y$, for a many-body ground state of a nontrivial QSHI. The subspace of occupied bands in the physical spectrum is not closed under time reversal. (b), (d), and (f) label the new functions after $\pi$ charge and $\pi$ $\Gamma$ fluxes are inserted in succession through the symmetry axis of the cylinder. For convenience, bulk bands away from the gaps are portrayed as dispersionless. Eigenvalues that are colored red, green, and blue correspond to eigenstates that extend over the left edge, the entanglement cut, and the right edge of the cylinder, respectively. Kramers’ doublets at $k_y=0$ and $\pi$ are labeled by thick dashes in the physical and entanglement energy spectra; occupied single-particle states of a many-body Slater determinant are labeled by black circles.

Figure 9

(Color online) Trace of $G^A$ vs flux $\Phi$ for a trivial ${\chi }^{Z_2}$-even (red) and a nontrivial ${\chi }^{Z_2}$-odd (blue) insulators. Flux is threaded by inserting a matrix $e^{i2\pi \Gamma \Phi /{\Phi }_o}$ at nearest-neighbor hoppings between $y=10$ to 1 on a $10\times 10$ lattice. Our Hamiltonian is Eq. (22) with (i) an additional inversion-symmetry-breaking term $0.1\sin k_x{\sigma }_1\otimes I_o+0.1\sin k_y{\sigma }_2\otimes {\tau }_3$ that couples spin-up and spin-down sectors and (ii) a weak random on-site potential so that the insulator is disordered; ${\chi }^{Z_2}$ is odd (even) when $m=1$ ($-1$). In (b), we choose $\Gamma ={\sigma }_1\otimes I_o$ ($I_s\otimes {\tau }_2$). For both choices of $\Gamma$, the trace of $G^A$ smoothly changes by 1 as $\pi$ $\Gamma$ flux is threaded through the nontrivial insulator. In the trivial case, the traces at $\pi$ and 0 fluxes are equal.